Topsy-Turvy Instant Runoff

Instant runoff voting (IRV) is known to display a number of perverse problems when there are more than two strong candidates. As a thought experiment, and mental exercise, Professor Warren Smith--mathematician and founder of the Center for Range Voting--try to determine the simplest IRV election that can showcase the greatest number of these issues.

9 voters: A > B > C
12 voters: B > C > A
8 voters: C > A > B

The example has only three candidates, and needs just 29 voters, but still illustrates:

• Reversal Paradox: if all the ballots have their order reversed, the "winner" stays the same, so the "best" choice is also the worst choice. (It's A in both cases.)
• Less-is-More: one of two forms of non-monotonicity, where by lowering the rankings for a losing candidate on certain ballots, they become the winning candidate. (Change two B-first voters to C-first, and B wins instead of A.)
• More-is-Less: the other form of non-monotonicity, where raising the winning candidate's rank on certain ballots makes them lose. (Raise A from bottom to top on 5 of the B-first ballots, and C wins instead of A.)
• Participation Paradox: where, if certain voters had just stayed home instead of voting, the result of the election would have been better for them. (Remove 5 B-first voters (who rank A last!) and C wins instead of A.)
• Anti-Participation Paradox: where, if more of a certain group of voters had shown up and voted, the results of the election would have been worse for them. (Add 2 C-first voters (who rank B last!) and B wins instead of A.)
• Precinct Paradox: the ballots can be divided into precincts, and the winner in each precinct is the same, but is not the same as the winner of the overall election. (Break into 3 precincts such that A-first/B-first/C-first in each is 3/4/4, 3/4/4, 3/4/0; B wins in each precinct, but A wins the aggregate.)
• Tactical Opportunity: some voters could prevent their least-favorite choice from winning by ranking a different candidate above their honest favorite. (If at least 2 B-first voters instead rank C above B, then A doesn't win; either B or C wins.)

I want to emphasize the last point, tactical opportunity; because this is the number-one problem that I see IRV-proponents falsely claiming that IRV doesn't have. It is simply not true that IRV removes the incentive to vote for the "lesser of two evils". There are less-contrived examples that can show this too, but it's worth pointing out every time it comes up.

How would approval voting handle this election? That, of course, depends on exactly how many voters in each faction approve of their second choice, but the Nash equilibrium is a win for B. I believe this is also true for IRV, but it involves the majority of voters ranking their second-favorite above their true favorite. Meanwhile, under approval it never helps you to lower your vote for your true favorite. I'll explore equilibria more in a future post.

Non-Monotonicity: Part II

In Part I, we defined non-monotonicity, showed that Instant Runoff Voting is a non-monotonic voting system, and that the recent mayoral election in Burlington Vermont suffered from non-monotonicity.

Here in Part II, I'll talk about why monotonicity matters, and do my best to refute all arguments to the contrary.

What happened?

To understand why it's important, first we should try to understand what happen to make an IRV election display non-monotonicity; we'll use what happened in Burlington's IRV election to showcase. I mentioned that "monotonicity" is an idea borrowed from algebra, but let's make that more clear: rather than move just 8 votes from Wright to Kiss, as we did in the example in Part I, let's make a function out of it, so we can see what it's like to move any number of votes over, and graph it against the vote-tallies in the final IRV round. A picture should make this clear:

You can see the huge discontinuity at 8 votes, where the IRV algorithm changes which candidate is eliminated. You can also see that, at 14 votes, the election switches back into Kiss' favor. So there's a limited window where this sort of manipulation will work. Foremost, obviously, is that there must be at least three candidates in the election. As candidates are eliminated, when only three are left, they must each have more than 25% of the vote. Finally, the eliminated candidate (the one with the fewest first-place votes) must be the "beats-all" (or "Condorcet winner") candidate, meaning that, in any one-on-one race, they would win. We can determine this by, for each pair of candidates, removing all but that pair from every ballot, and seeing who would win. For the Burlington election, check, check, and check.

Under these circumstances, IRV exhibits non-monotonic behavior, and eliminates the Condorcet winner. In fact, you could say that because it eliminates the Condorcet winner, it has exhibited non-monotonic behavior. But so what? Well, there's a hidden implication in the elimination of a Condorcet winner, and that is that the Condorcet winner is the moderate candidate, the compromise candidate. Meaning that, by eliminating them, we've chosen an extremist candidate.

In the Burlington election, in addition to the nationally-familiar Democratic party (with candidate Andy Montroll) and Republican party (with candidate Kurt Wright), we also had the eventual IRV-winner, Bob Kiss of the Vermont Progressive Party, which is just a bit more left-leaning than the Democrats. There is a model of voting behavior, appropriately refered to as the "1D" model, where candidates and voters are arrayed on a line, left to right, and voters always more-prefer the candidate closer to them. In the 1D model then, we should expect that all Kiss-first or Wright-first voters would have had Montroll second (ignoring, for now, candidates other than these three). And that's almost the case: among those who expressed a preference, Wright voters prefered Montroll 3:1 over Kiss, and Kiss voters prefered Montrol 7:1 over Wright. 1D isnt' perfect, but it's often pretty close (how much of this is feedback from our two-party dominated political process is an article for another day), and in my "simplified" example of the election, I moved all the "wrong" votes to their "proper" 1D vote. If the purpose of an election is to choose an appropriate candidate to represent a body of people, it's logical to say that you want the winner to come from the center of the spread of voters' opinions; it's a compromise.

But IRV doesn't always pick the compromise candidate; instead, it picks one of the two extremes. The Center for Range Voting has some excellent graphical representations of this, using a 2D political spectrum and many more candidates (which is more realistic than our 1D, 3 candidate example). At a glance you can see how extremist candidates encroach on the territory of more moderate candidates.

On one hand, it's very strange for me to be damning this election, since it technically elected a third-party candidate, which I generally find to be a good thing. One of the reasons I find our current system so abhorrent is because third-party candidates actually damage their most closely-aligned major-party by acting as spoiler candidates. On the other hand, that's what happened in Burlington. Nearly two-thirds of Burlington residents are registered Democrats (click "voting" on that page to expose the data). And considering the Democratic wave that has been washing over the nation recently, in a vote between right, left, and lefter, the Republicans are the "third" party in Burlington. And as a third party, they operated as we would expect; they siphoned votes away from their nearest major-party candidate and gave the election to their most-hated opponent. Their nearest major-party candidate just happened to be the Democrats.

The "third party" Republicans opperated as a spoiler. Under the 1D model (and by a 7:1 margin in actuality) they would prefer the more-moderate Democrat, Montrol, over the further-left Progressive, Kiss. But their votes for Wright acted only as votes against Montroll in the final tally. This is exactly analogous to Nader spoiling Gore. Despite IRV proponents claims, IRV does not eliminate the effect of spoilers, it only delays their effect. Under our current plurality system, spoilers can have an effect with as little as one vote, if the other candidates are close enough. Under IRV, they only pop up when they have at least 25% of the vote. But the effect will ultimately be the same: fear of spoilers will tend to drive voters back towards the two major parties.

Both of these things—failure to elect a "beats all" winner and susceptibility to a "spoiler" candidate—are indicated by IRV's non-monotonic behavior. Non-monotonicity is the smoke to their fire, or the chink in the armor that lets them sneak in. As IRV advocates will claim, it doesn't matter that if some voters changed their votes they could change the outcome in a non-intuitive way; because they didn't. It matters because that possibility indicates that elections will tend toward extremism and towards two-party domination.

How often?

After being forced to admit that, yes, non-monotonicity is something to be concerned about, an IRV advocate's second round of defense will be that it doesn't matter because it doesn't happen that often. And I must admit, this one example is just one example; the plural of "anecdote" is not "data". Luckily, a host of very smart individuals have tried to calculate how often these things should happen, which CRV has taken the time to summarize. The first to make the attempt was Crispin Allard in 1996, and he calculated the chance as 1 in 40,000, and this is the number upon which IRV advocates have built their argument. Unfortunately, Allard's calculation was wrong. Very wrong. He only counted the more common of the two types of non-monotonicity, and he took advantage of a six-fold symmetry to simplify his calculations but forgot to multiply by six at the end, but worst of all made a serious arithmetic error. Over all, he was off by a factor of over 1,000.

Corrected calculations have found that, using Allard's model and his assumptions, the correct odds of an IRV election displaying non-monotonicity is closer to 1 in 7.

But, an IRV advocate might say, that's only one model. Yes, it is. Using a different model, the odds were calculated to be around 1 in 18, and in a third, a little less than 1 in 10. That, combined with the growing list of real-life IRV elections that have displayed non-monotonicity, it should be quite clear that these failures are predicted to happen often, and often happen in real life. Also, it's important to note that these values apply only when there are three strong candidates in a race. As the number of viable candidates increases, the odds only get worse, until the chaos is practically guaranteed.

What's it mean?

Ultimately, adoption of IRV will result in practically no change to the status-quo. Third parties will find they can gather more votes, but they still won't be able to win any elections, because of the fear of spoilers. Voters will still find themselves deciding between the least-evil of a pair of party-chosen politicians, and we'll continue to lurch back-and-forth between left-leaning policy and right-leaning policy, rather than pick a stable course. The divide between "red" and "blue" will continue to grow and our common ground will continue to wither away. And let's not ignore that this non-improvement comes at a high cost (another post for another day). Why not focus on a change that will actually change something?

I've come down hard on IRV here, but that's only because it's the loudest competitor. The truth is, every election method that uses ranked-order ballots (i.e., ballots of the type A > B > C) is susceptible to one of two failings, either "favorite-betrayal" (which IRV displays through its non-monotonicity) or something called "cloning" (which, briefly, involves proping up ideological copies of your opponent in order to split the votes of those who oppose you). IRV is immune to cloning, a fact that they proudly showcase in this video, while ignoring their non-monotonicity pitfall. ("Or something like that" indeed!) Now, cloning is a serious concern under our current plurality system; but why fix one problem by introducing another, when score voting is clone-proof and monotonic, not to mention cheaper to implement?

Non-Monotonicity: Part I

Since the results of Burlington's mayoral election were announced, battle has raged across the internet between advocates of instant runoff voting and those who have better ideas. Several points are being argued discussed, but the most confusing in the issue of non-monotonicity. What's that word even mean?

Monotonic, if you remember your algebra, means that a function, f(x), is either always increasing or always decreasing as x increases. A function that goes up for a little while and then goes back down is non-monotonic. You could think of it as "if more is better, than even more is even better". And that's (sort of) how the analogy comes over to voting methods: under a monotonic method, getting more votes is always better. So under a non-monotonic method, getting more votes is sometimes bad, or, getting fewer votes is sometimes good.

There are three issues at stake here: first, is IRV non-monotonic? This is an unqualified "yes", although IRV advocates will claim that it's so rare as to be inconsequential in real-life elections. (More on that later.) Second, did the recent election in Burlington exhibit non-monotonicity? This is also an unqualified "yes", which one might think would give pause to those claiming that non-monotonicity is ultra-rare. (Rather, they have decided to argue that it didn't really happen.) And finally, does it matter? This is the only issue of the three which can be up for debate, which we'll do in Part II. But first, we'll pause to prove points one and two.

IRV is non-monotonic

The easiest way to show this, is to construct an example. First, the election needs to have at least three candidates; let's call them 'K', 'M', and 'W'. Secondly, we need to construct a set of ballots where one candidate, let's pick K, is the winner, but in such a way that if we improve K's ranking on some of the ballots, they instead lose. Easy enough:

29 ballots: K > M > W
25 ballots: M > K > W
32 ballots: W > M > K

M, with the fewest top-ranking ballots, is eliminated, and K defeats W by an impressive 54 to 32 margin. But! if we improve K's position on just eight ballots, like this:

29 ballots: K > M > W
25 ballots: M > K > W
24 ballots: W > M > K
8 ballots: K > W > M

Then instead candidate W, with only 24 top-ranking ballots, is eliminated, and M defeats K by a 49 to 37 margin. To repeat: by improving K's score on a small number of ballots, K went from being the winner to being the loser. Which is a very non-intuitive result. It's also possible to create a situation where decreasing a losing candidate's rank on some ballots can cause them to win (but I leave that as an exercise to the reader). If either of these cases is possible to construct for a given voting method, then getting more votes is not always better, and by definition the voting method is non-monotonic.

What about Burlington?

So clearly these situations can happen under IRV. The next question is if the Burlington election was one of those situations. The proof of this is almost trivial, since the numbers (and names) in the above example were pulled from the results of the Burlington election (with a few complicating but non-relevant omissions.)

The real question is, why do IRV's supporters insist that this is not the case? I wish I knew! It seems to stem from a lack of understanding of the words "if" and "could", as in "If the outcome of the election could be changed by...", as their argument is that this election "could" have been non-monotonic "if" some voters changed their ballots "but they didn't". No, no, no! If they could then it is non-monotonic! A lot of bad-blood is being spilled over this non-disputable point, and I don't understand why the opposition persists, as it only serves to distract us from the real discussion.

Which we'll cover in the next installment.